I don’t like the mortgage calculators banks put on their website. You type in a bunch of numbers and then you get one number back.

Life moves much faster than this. Now you think it’s 30 years, then it’s 25, mmmh, and how about an interest rate of 5.5% p.a. instead of 4.8% p.a. You don’t want to type that in each time, do you?

These mortgage calculators are also not showing all the information. Rarely do you see straight away how much interest you have to pay. For a 30-year loan that can be as much as the principal loan you want to borrow.

Worst of all, you wouldn’t learn how amortization works. And that would be a shame.

Technical details

This is for those who want to see what’s going on under the hood.

Monthly repayments are conditional on the length of the loan term (horizontal axis) and annual interest rate (vertical axis). Click on any point in the graph and the sliders above for Loan term and Annual interest rate will be updated.

Have a go and see what happens when you move left and right or up and down. Pay attention to the Total interest. For example, if you try to follow one of the contour lines, say $4,000, from lower left to upper right, that means you’re still paying the same amount each month. But the the total paid depends on where you are on that $4,000 contour line.

Now, imagine you have a much higher loan and the loan term extends to infinity. This way you could repay a higher loan indefinitely with a reasonable monthly repayment. That’s what we call rent. 🤩

How it works

The contour lines show combinations of interest rate and loan term that produce the same monthly repayment for the selected loan amount. This makes the trade-off visible: for a fixed loan, a shorter term can have the same repayment effect as a higher interest rate, and vice versa.

The maths behind

The equation for a fixed scheduled repayment \(M\) of a principal loan \(P\) over a number of periods \(n\) with an interest rate \(r\) is

In case of monthly repayments, we divide the annual interest rate (it’s usually given as % p.a., which is short for per annum, which is Latin for per year) by 12 and have to multiply \(n\) with 12 if the duration of the term is in years.

Let’s find out under what conditions the principal equals the interest paid.

The total amount repaid is \(n\,M\). The total interest is \(I = n\,M - P\). And we want:

A little algebra turns this into

or:

We can substitute the amortization formula back for \(M\):

\(P\) cancels out. This means our answer doesn’t depend on the principal loan amount \(P\).

We are left with

We can rewrite \(r\) and \(n\) in annual-rates-and-years form. With \(R\) as the annual interest rate and \(T\) as term in years we have \(r = \frac{R}{12}\) and \(n=12\,T\). So

The exact condition becomes

The quality of principal and interest depends mostly on the product \(R\,T\), or the annual rate \(\times\) years.

We can get a useful heuristic when we replace monthly compounding with annual compounding

Then the condition becomes

Let \(x = R\,T\), then

This equation has the solution \(x \approx 1.594\). So the mental rule is \(\boxed{\text{annual rate}\;\times\;\text{years} \approx 1.6}\).

Examples:

So whenever you find a loan in the wild where the product of annual interest rate times its duration equals 1.6 you know that someone is paying the same amount on interest as their loan is worth.

Investment?

Hi there. If you’re like me, you feel like you want to kick yourself in the butt for not having invested into your own future. Life, when you’re young, just feels too good to think about any of that. And do you really have any extra cash that you could save? $50. Would it even matter?

It’s never too late. So why not start today?

The big idea is of course compounding. There are many great shortcomings of the human race, but according to physicist Dr. Albert Bartlett:

The greatest shortcoming of the human race is our inability to understand the exponential function.

Investment Scenario Calculator

Let’s see how far you can get with saving a little extra every year (10%).

Here are a few sliders to play around with. There are dollar amounts (I live in New Zealand, so don’t read too much into the numbers. 1 NZD today is about USD 0.58, or EUR 0.50) and there are rates. And there are years (otherwise there wouldn’t be much compounding, would there?)

You can start with zero capital, or with no savings and let only your invested capital compound.

If you know by how much your annual income would grow, you can do change that rate, too.

I’ve also included Uncertainty. It increases or decreases the rates by that much, so we can check out worst and best case scenarios. (The numbers in brackets show the values for worst- and best-case).

There is a line called Retirement Income. This is an inflation-adjusted annual lump sum you would receive after tax, if you were to move all your wealth into a term deposit and just live off the interest rates.

The last two lines are sensitivities. I talk about them down below.

OK. I’ll leave you to it. Happy investing!!

Reset

Technical Details

This is the basic formula for everything above.

The nominal captial is quite large. That’s because we haven’t adjusted it for inflation. Dividing the nominal capital by the total inflation over N years, gives us the real capital, in terms of today’s purchasing power.

I was also curious about how tiny adjustments in behaviour might lead to bigger savings. There are two things we can control: savings rate and extra years.

Mathematically, the sensitivities can be calculated using derivates. Taking the derivative of \(K_{nom}\) (or \(K_{real}\)) with respect to \(s\) gives us the sensitivity to changing the savings rate.

You’ve probably seen that some sliders don’t make a difference in the +1% Savings Impact. Now, you can look for yourself why that is. (Yes. LOOK AT THE FORMULA!!!)

Astonishingly, the savings rate itself doesn’t make a difference. Why? Because every additional 1 percentage point of savings buys you the same chunk of lifetime wealth, regardless of whether you’re already saving 2% or 40%.

In real life, you also don’t have much control about \(Y_1\), \(r\), \(g\); all variables that determine the sensitivity. But you have control of \(N\), the time you let your wealth accumulate.

Let’s look at the sensitivity of saving for another year.

I know, there is a lot going on here. But we can break it up and look into the two terms of the above sum.

The first term is related to your starting capital compounding over the years (\(K_0 (1+r)^N\)) and it will simply grow some more for another year. If you dial the slider of your year-1 income down to 0, you can see that the +1 Year Impact is the Real Capital times the Investment Return Rate. An extra year is valuable because it applies your return rate to your entire accumulated financial history.

The second term is related to your savings. It’s a bit too complicated to explain it in simple terms (and there is a minus in the numerator of the fraction). But you can think of it as growth on past savings (\((1+r)^N \ln(1+r)\)) that is adjusted for income growth (\(-(1+g)^N \ln(1+g)\)).

To sum up all of the details about the sensitivities, just keep in mind

Savings rate controls how much fuel you add, but time controls how long it burns.

Notan is a Japanese word that is used in arts to describe the balance of light and shade. As a visual aid, Notan helps us seeing an image as value shapes, not as contour lines.

This tool is insipred by the Proko’s values tool.

Blur: Levels: Download PNG

They can be fun.

They can be hard.

And they can be educational.

The maze (generator) below is using Kruskal’s Algorithm and it is inspired by Professor L.’s Maze Generation Algorithms - An Exploration. You should check out her other mazes generation algorithms and corresponding explainers.

Size:

Generate Maze

Save as PNG

As a motivating example, I showed an aerial image that I took during a helicopter flight aorund Aoraki / Mount Cook National Park in May.

There is so much structure in that image and I thought, “I could come up with an approximation for that river bed!”.

A few hours later (and thanks to ChatGPT), I had a phenomenological model that would describe the main features of the river bed and the flow of the water within.

The toy model below has 3 main parameters, displacement, decay, and max dimension. You can try to match the real world by changing those. water percentage is a visual effect that color-codes a percentage of all lines in a watery color.

Seed: Randomize Seed

Displacement 0.3

Decay 0.8

Max Dimension 6

Water Percentage 0.05

Further details

Rivers naturally form curves, or meanders, because of the way water flows and interacts with the river banks. When water flows around a slight bend, it moves faster on the outer edge and slower on the inner edge. The faster water erodes the outer bank, while the slower water deposits sediment on the inner bank. This feedback makes the bend grow over time, producing the winding patterns we see in meandering rivers, oxbow lakes, and floodplains.

In our simulation, we mimic this process by repeatedly shifting the midpoint of each river segment in a random perpendicular direction. This models the instability and sideways migration of real rivers. Mathematically, the river’s centerline \(y(x)\) could be described by a simple equation:

Here, \(D\) represents smoothing effects like natural sediment spreading, and \(\eta(x,t)\) is a random term representing local variations in erosion and deposition.

The midpoint displacement algorithm is a simple way to recreate this: each subdivision adds a small random shift, controlled by a displacement parameter \(p\) that gets smaller at finer scales (controlled by decay \(d\)). After that, Chaikin’s smoothing spreads out sharp bends, similar to how real rivers naturally diffuse their curves. Together, these steps create winding, natural-looking river shapes that balance the randomness of erosion with the smoothing effects of sediment redistribution.

I am going to create an intuitive understanding of Bayes’ Theorem, or Bayesian Inference. I won’t go into the technical details because, that’s what many intros to Bayesian inference are doing. Instead, we play and learn.

I will still give you Bayes’ Theorem, so you can try to see how it works it’s magic in the coin flipping example below. All I’m saying is that Bayesian inference is counting.

The term on the left \(p(\theta|x)\) is the posterior - How probable is the parameter \(\theta\) given the data \(x\).

The terms on the right are the likelihood \(p(x|\theta)\) - How probable are the data \(x\) given the parameter \(\theta\)?

The prior \(p(\theta)\) - How probable is the parameter \(\theta\) without seeing any data \(x\)?.

And, lastly, the evidence \(p(x)\) - How probable is it to observe the data \(x\)?

Now, let’s forget that this means anything. The only thing we need to know is that when tails (T) comes up, the probability distribution function \(p(T) = 2-2x\) and \(x \in {0,1}\). That’s a linear function that is 2 at \(x=0\) and 1 at \(x=1\). It’s also a right triangle and a proper probability distribution function because \((2\times1)/2 = 1\) is the right triangle’s area, which is also the area under the curve. You can see it by typing in T.

Now, for heads, it’s similar. The probability of H is \(p(H) = 2x\). That’s a linear function that is 1 at \(x=0\) and 2 at \(x=1\). Again, it’s both a right triangle and a proper probability distribution function. You can see it by typing in H.

OK. What happens next feels like magic: When you enter multiple coin flip results (e.g., TT, or HT, or HTHHT...), the probability turns into a more recognizable probability distribution. Wow! Why is that? It is simple (and repeated) multiplication of those triangles. Really! (You can check the source code of this site and look for f_head(x) and f_tail(x).

So, what’s going on? We start from the prior belief that both heads and tails are equally likely: a uniform prior distribution. If we observe tails, we multiply our prior with the “tails triangle”. This now becomes our new prior and with each subsequent coin flip, we keep updating our posterior based on previous evidence.

That’s what I mean by counting. You just multiply it by what you see (kinda).

(I haven’t talked about evidence \(p(x)\) and I won’t.)

So have a go yourself. Below is an input form where you can enter your coin flips. Starting form a uniform distribution of heads (H, on the right) and tails (T, on the left), you can see how the posterior is updated based on the new data coming in, i.e., the evidence.

Enter sequence of H and T:

a data generating process is a process in the real world that “generates” the data one is interested in.

Thinking about the data generating process has been a game changer for me.

Before getting into the weeds, let’s unpack what data generating process means and how you can can use it as a mental model. If we flip data generating process around, we get the sentence:

Processes generate data.

1. Process is “a series of actions, motions, or occurrences”.
2. Generate means “to bring to life”.
3. Data (plural for datum) are “facts or principles granted”

I like those old dictionary definitions. Webster’s 1913 is particularly rewarding in its essence and clarity. Using our new words (or old?), let’s create a more vivid description of data generating process:

A series of actions, motions, or occurrences bring to life facts.

Splendid!

Isn’t that how we think the world keeps going? I think so, too: Something is going on, and “poof!”, there’s an outcome, and when we look at it we try to make sense about what was causing the outcome. What is that Something? And that Something is the (data generating) process.

Let’s say you are looking at a series of data points, for example, a time series. Depending on the domain, you might describe what you see as follows:

• “The numbers go up and down a lot, but over time you can see which way things are moving.” (Election Polls)
• “There’s a general upward trend, but with occasional periods of decline.” (Stock Market)
• “Over time, things are getting warmer, though there are still some cooler periods.” (Climate Change)
• “The trend is upward, but sales can vary a lot from month to month.” (Retail Sales)

Fact number one: Looking at data makes you think about the What.

That’s good but not good enough. You can do better. You want to understand “Why” things are the way they are and “How” they came to be.¹ You are describing that facts but you are not (yet) looking into what kinds of actions have brought those facts to life. If you were to think about actions, motions, or occurrences, you start thinking about what those actions, motions, or occurences would give rise to.

I look at data a lot. Being a climate data scientist, I am curious about the past, the present, and the future of the climate system. Climate science, as a discipline, is rich in data. There are about 17,500 stations around the world that record weather data (source). There are currently 322 earth observation satellites in orbit around Earth (source). We do that because

Accurate weather forecast and climate prediction are crucial for decision making and support for appropriate action to mitigate the impacts of natural hazards and climate change. - WMO

Weather forecast goes beyond looking at data. And that brings us to fact number two.

Fact number two: Generating data makes you think about the How.

Here, by data I mean fake data². Yes, I am saying that you should create fake data. Everyone should. You might actually learn something. Creating fake data is not easy. Particularly, if you want to make your fake data look real. Here’s a challenge.

Try to fake the US stock market data for the last 50 years!

Or, take Global Temperature. Try faking that.

A brief history of weather forecasting

Let’s look at weather forecast again because it’s a prime example of how we got from looking at data to making predictions (about the weather). According to history, the Greeks where one of the first meteorologists, and Aristotle is considered the founder of meteorology, having written his “Meteorological” around 340 B.C. Skipping ahed 2200 years and we are witnessing the birth of modern weather forecasting in the 1860s.

Today, numeral weather prediction models are tools of the trade for meteorologists and climate scientists. Weather forecasts have become much more accurate: “A four-day forecast today is as accurate as a one-day forecast 30 years ago.” We all use weather forecast data on a daily basis on our phones. Open your weather app and you are likely to find some information about the data. If I scroll down on my iPhone weather app, for example, I can find a link to weather data. At the bottom of the page, you will find all the data sources the iPhone weather uses. (All of this is fake data, but in the best and most useful way imaginable.)

By the way, predicting the weather is way easier than predicting the stock market. But that’s another topic.

Now, how can you actually generate data? In practice, you can run numerical simulations of the thing you are interested in. That is usually some sort of computer program that will generate some data outputs for you.

Weather forecasting itself is too complicated for this post, so I’ll guide you through a simpler, yet, as insightful, example.

The coin toss

One of my favourite examples for everything is the coin toss. Everyone has a coin and can flip it. It’s a real-world process and one that is simple enough to help our understanding.

Depending how many times you flip the coin you generate observations, or data, from that process. You guessed it, the coin toss itself is a data generating process.

Here’s a coin toss example in Python. (You can copy it or download it here.)

import random

def coin_toss(num_tosses,p_head):
    return random.choices(["H", "T"], weights=[p_head, 1-p_head], k=num_tosses)

if __name__ == "__main__":
    num_tosses = int(input("Number of coin tosses: "))
    p_head = float(input("Probability of tossing heads (between 0 and 1, default=0.5): ") or "0.5")
    tosses = coin_toss(num_tosses, p_head)
    print(f"Results after {num_tosses} tosses:")
    print(", ".join(tosses))
    heads_count = tosses.count("H")
    print(f"Heads: {heads_count}")
    tails_count = tosses.count("T")
    print(f"Tails: {tails_count}")

The method coin_toss() generates our facts which we record in the variable tosses. My output from tossing 10 times using the default probability of 0.5 is

Number of coin tosses: 10
Probability of tossing heads (between 0 and 1, default=0.5): 
Results after 10 tosses:
H, H, T, T, H, T, T, T, H, T
Heads: 4
Tails: 6

Now that we have a model of our coin toss world, we can create different outcomes every time we run the code example. We can pick probabilities that are different from a fair coin (p=0.5). We can repeat the coin toss but for more tosses. Here’s a thousand, with a probability of heads of 0.2.

Number of coin tosses: 1000
Probability of tossing heads (between 0 and 1, default=0.5): 0.2
Results after 1000 tosses:
H, H, T, T, T, T, T, T, H, T, H, T, T, H, T, T, H, T, T, T, T, T, T, H, T, T, H, T, T, H, H, T, T, T, T, T, T, H, H, T, T, T, T, T, T, T, T, T, T, T, H, T, T, H, T, T, T, T, T, T, H, T, T, T, T, T, T, T, T, T, T, T, H, T, T, H, T, T, T, T, T, T, T, T, T, H, T, T, H, H, T, T, T, T, T, T, H, T, H, T, H, H, T, H, T, T, T, T, T, T, T, T, T, H, T, T, T, H, T, T, H, T, T, T, T, H, H, T, T, T, T, T, T, T, T, H, T, H, T, T, T, T, T, H, T, T, H, T, H, T, H, T, T, T, T, T, H, T, T, H, T, H, T, T, T, T, H, T, T, H, T, T, T, T, T, H, T, T, T, T, H, T, T, T, T, T, T, H, T, T, T, T, H, T, T, T, T, H, T, T, H, T, T, T, T, T, H, H, T, T, H, T, T, T, T, H, T, T, H, T, T, T, T, T, T, T, T, T, T, T, H, T, T, T, T, T, T, T, H, H, T, H, H, T, T, T, T, T, T, T, T, H, T, T, T, T, T, T, T, T, T, T, T, T, T, H, T, T, T, T, T, T, T, T, T, T, H, H, T, T, T, T, T, T, T, T, T, T, T, H, T, T, T, H, H, T, T, H, T, T, T, T, T, T, T, H, T, T, T, T, H, T, T, T, T, H, T, T, T, T, T, H, T, T, T, T, T, T, H, T, T, T, T, T, T, T, T, T, T, T, H, T, T, H, T, T, T, T, T, T, H, T, T, H, T, T, T, T, T, T, T, T, H, T, H, T, T, T, T, H, T, T, T, T, T, T, T, T, H, T, T, T, T, T, H, H, T, T, H, T, H, T, H, T, T, T, T, H, H, H, T, T, T, H, T, T, T, H, T, T, T, T, T, T, H, T, T, H, T, T, H, T, T, T, T, T, H, T, H, T, T, H, T, T, T, T, T, T, T, H, H, T, T, T, T, T, T, T, T, T, T, T, T, T, H, H, T, T, T, T, T, T, H, T, H, H, T, T, T, T, T, H, T, H, H, T, H, T, T, T, T, T, T, T, T, H, T, T, T, H, H, T, T, T, T, T, H, T, T, T, T, T, T, T, T, T, H, H, T, T, T, H, H, T, T, T, T, T, T, H, H, T, H, H, T, H, T, H, T, H, T, H, T, H, T, T, T, H, T, T, T, T, T, H, T, H, T, T, T, T, T, H, H, H, T, T, H, T, T, T, H, T, T, T, T, T, T, T, T, T, H, H, H, H, H, H, H, T, T, T, T, H, T, T, H, T, T, T, T, T, H, T, T, T, H, T, H, T, T, T, T, H, T, T, T, T, T, T, T, T, T, T, T, T, H, T, T, T, T, T, T, T, T, T, T, T, T, T, T, H, H, T, T, H, T, T, T, T, T, H, T, T, T, T, T, T, T, H, T, T, H, T, T, T, H, H, T, T, T, T, T, T, T, T, T, H, T, T, H, T, T, T, H, T, T, H, T, T, T, T, T, T, T, T, H, T, T, H, T, H, T, H, T, H, T, T, T, H, T, T, T, H, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, H, T, T, T, T, T, T, T, T, T, T, T, H, T, T, T, T, T, T, T, T, H, T, T, T, T, T, T, T, T, H, H, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, H, T, T, T, T, T, H, T, T, T, T, T, T, H, T, H, T, H, T, T, H, H, T, T, T, T, H, T, T, T, T, T, T, T, T, H, T, H, H, H, T, T, T, T, H, T, T, H, T, T, H, T, T, H, T, T, T, T, H, T, T, H, T, T, H, T, T, H, T, H, T, T, T, T, T, T, T, T, T, T, T, T, T, T, T, H, T, T, T, T, T, T, T, H, T, H, T, T, T, H, T, H, H, T, T, T, T, T, H, T, T, T, T, T, T, T, H, H, T, T, T, T, T, H, T, T, T, T, T, T, H, T, T, T, T, T, T, T, T, T, T, H, H, T, T, T, T, T, H, T, T, T, H, H, T, H, T, T, T, H, T, T, T, T, H, T, H, T, T, T, T, T, T, T, T, T, H, T, H, T, T, T, T, T, T, T, T, T, T, T, T, H, H, T, T, T, T, T, T, T, T, T, T, T, H, T, T, T, T, H, T, T, T, T, H, T, T, T, T, H, H, T, T, T, T, T, T, T
Heads: 218
Tails: 782

We can see that we don’t always get the exact share of heads and tails (200 H out of 1000 tosses) and that’s ok. Tossing a coin, as we know from the real world creates different outcomes every time we do it. That’s why in football

the referee tosses a coin and the team that wins the toss decides which goal to attack in the first half or to take the kick-off. - Law 8: The Start and Restart of Play

That’s why our model contains randomness in the first place (random.choices(["H", "T"],...). We know that coin tossing looks random, so our model must reflect the fact.

Fact number three: Examining your model lets you figure out the Why.

That’s it. The data generating process.

What I like about jupyter books, in particular, is that you can integrate your code snippets and the output will be automatically rendered as HTML content. A nice little side effect of it is that you will know whether your code works or not. A little bit like unit testing in software development.

jupyter {book} feature list:

• Support for Markdown files, Jupyer notebooks, and MyST Markdown notebooks (Markdown files that will be converted to a notebook and excecuted 🤩)
• Preview your content
• Build a PDF
• Citations and bibliographies
• Math and equation support
• Interactive data visualizations

What I haven’t figured out, yet, is how to integrate the HTML content into Jekyll, a framework that GitHub Pages uses for turning Markdown files into a website (like this one). I would like to have that content on a subdomain.

Wait and see.

Pink noise:

Reproducing Figures 1 and 2 from Vaughan¹ et al. (2011):

Use docker to deploy and run locally:

docker-compose -f docker-compose.yml up --build

It used to run on Heroku’s free plan, which they have removed since then ¯\(ツ)/¯.

Update:

It’s up and running again on render.com